The first nonautonomous version of the Grobman-Hartman theorem was established by Palmer [1]} for the case of continuous time dynamics. A discrete time version of Palmer's result was formulated in [2]}. Both these results were obtained by assuming that the associated linear system admits a uniform exponential dichotomy.
The problem of smooth linearization for nonautonomous systems was considered only recently. To the best of our knowledge, the ﬁrst results in this direction
were obtained in [3]}, [4]}, in the uniformly/nonuniformly exponentially stable case. Moreover, in [5]} the authors have established a Sternberg-type theorem in the setting of
a uniform exponential dichotomy with continuous time.
More relevant to our context are the works [6]}, [7]} (see also [8]}) where \(C^1\) , differentiable (at 0) and Hölder linearization results were obtained, respectively, in the setting of a nonuniform exponential dichotomy under some spectral gap and spectral band conditions. These results will play a major part in our proofs, as we already emphasized.
